Answer
$$\eqalign{
& \left( {\text{a}} \right)x = - 2 \cr
& \left( {\text{b}} \right){\text{Decreasing on: }}\left( { - \infty , - 2} \right),{\text{ Increasing on: }}\left( { - 2,\infty } \right) \cr
& \left( {\text{c}} \right){\text{Relative minimum at }}\left( { - 2,0} \right) \cr} $$
Work Step by Step
$$\eqalign{
& {\text{We have }}f\left( x \right) = {\left( {x + 2} \right)^{2/3}} \cr
& \left( {\text{a}} \right){\text{Calculating the first derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {x + 2} \right)}^{2/3}}} \right] \cr
& f'\left( x \right) = \frac{2}{3}{\left( {x + 2} \right)^{ - 1/3}} \cr
& f'\left( x \right) = \frac{2}{{3{{\left( {x + 2} \right)}^{1/3}}}} \cr
& {\text{There are no points at which }}f'\left( x \right) = 0 \cr
& {\text{The derivative is not defined at }}x + 2 = 0:{\text{ we obtain the critical }} \cr
& {\text{point }}x = - 2 \cr
& \cr
& \left( {\text{b}} \right) \cr
& {\text{Set the intervals }}\left( { - \infty , - 2} \right),\left( {2,\infty } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 180 }}} \right) \cr} $$
\[\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( { - \infty , - 2} \right)}&{\left( { - 2,\infty } \right)} \\
{{\text{Test Value}}}&{x = - 3}&{x = 0} \\
{{\text{Sign of }}f'\left( x \right)}&{{\text{ }}f'\left( { - 3} \right) < 0}&{{\text{ }}f'\left( 0 \right) > 0} \\
{{\text{Conclusion}}}&{{\text{Decreasing}}}&{{\text{Increasing}}}
\end{array}\]
$$\eqalign{
& \left( {\text{c}} \right) \cr
& {\text{By Theorem 3}}{\text{.6}} \cr
& f'\left( x \right){\text{ changes from negative to positive at }}x = 0,{\text{ so }}f\left( x \right) \cr
& {\text{has a relative minimum at }}\left( { - 2,f\left( { - 2} \right)} \right) \cr
& {\text{ }}f\left( { - 2} \right) = {\left( { - 2 + 2} \right)^{2/3}} \cr
& f\left( { - 2} \right) = 0 \cr
& {\text{Relative minimum at }}\left( { - 2,0} \right) \cr} $$
